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Geiger's Method (Definition)

Geiger's method [1] is an iterative procedure using Gauss-Newton optimization to determine the location of an earthquake, or seismic event. Originally his method was developed to obtain the origin time and Epicentre but it is easily extended to include the Focal Depth for Hypocentre determination.

Given a set of $M$ arrival times $t_i$ find the origin time $t_0$ and the hypocentre in cartesian coordinatios $(x_0,y_0,z_0)$ which minimize the objective function

$\displaystyle F(\mathbf{X})=\sum_{i=1}^{M}r_i^2.$ (1)
Here, $r_i$ is the difference between observed and calculated arrival times
$\displaystyle r_i=t_i-t_0-T_i,$ (2)
and the unknown parameter vector is
$\displaystyle \mathbf{X}=(t_0,x_0,y_0,z_0)^{\mathrm{T}}$ (3)
In matrix form (1) becomes
$\displaystyle F(\mathbf{X})=\mathbf{r}^{\mathrm{T}}\mathbf{r}$ (4)
The Gauss–Newton procedure requires an initial guess of the sought parameters, denoted here as
$\displaystyle \mathbf{X}^*=(t_0^*,x_0^*,y_0^*,z_0^*)^{\mathbf{T}},$ (5)
which are then used to calculate the adjustment vector
$\displaystyle \delta\mathbf{X}=(\delta t_0,\delta x_0,\delta y_0,\delta z_0)^{\mathrm{T}}$ (6)
in
$\displaystyle \mathbf{A}^{\mathrm{T}}\mathbf{A}\delta\mathbf{X}=-\mathbf{A}^{\mathrm{T}}\mathbf{r}.$ (7)
The Jacobian matrix $\mathbf{A}$ is defined as
$\displaystyle \mathbf{A}=\left( \begin{array}{cccc} \partial r_1/\partial t_0 &... ...& \partial r_M/\partial y_0 & \partial r_M/\partial z_0 \ \end{array}\right).$ (8)
The partial derivatives are evaluated at the initial guess, or trial vector, $\mathbf{X}^*$. Equation (7) can be rewritten as
$\displaystyle \mathbf{G}\delta\mathbf{X}=\mathbf{g}.$ (9)
Using (9) and an initial guess $\mathbf{X}^*$ an adjustment vector can be calculated. The initial guess can then be updated $\mathbf{X}^*+\delta \mathbf{X}$ and used as the inital guess in the next run of the algorithm. In this manner the sought parameters $\mathbf{X}$ can be determined to some tolerance.

Bibliography

1
Geiger, L., “Probability method for the determination of earthquake epicenters from the arrival time only.” Bull. St. Louis Univ. vol. 8, pp. 60-71.
2
Lee, W. H. K. and Stewart, S. W. Principles and Applications of Microearthquake Networks, Academic Press, New York. 1981
3
Gibowicz, S. J. and Kijko, A. An Introduction to Mining Seismology, Academic Press, New York. 1994.



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Keywords:  earthquake location, least squares, Gauss-Newton

Cross-references: algorithm, matrix, vector, parameter, function, Hypocentre, Focal Depth, Epicentre

This is version 2 of Geiger's Method, born on 2006-03-24, modified 2006-03-24.
Object id is 138, canonical name is GeigersMethod.
Accessed 2328 times total.

Classification:
Physics Classification: 91.30.Px (Phenomena related to earthquake prediction)
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